Lecture on Standard Deviation and Variance

Key Concepts

Standard Deviation

• Definition: A measure of how spread out numbers are in a data set.
• Population Standard Deviation: Used when you have the entire population data.
  • Formula:
    • ( \sigma = \sqrt{ \frac{ \sum{(x_i - \mu)^2} }{n} } )
    • (\sigma): Standard Deviation
    • (\mu): Population Mean
    • (n): Total number of data points
• Sample Standard Deviation: Used for a sample of the population.
  • Formula:
    • ( s = \sqrt{ \frac{ \sum{(x_i - \bar{x})^2} }{n-1} } )
    • (s): Sample Standard Deviation
    • (\bar{x}): Sample Mean

Calculation Example

• Data Sets:
  • Set 1: [4, 5, 6]
  • Set 2: [3, 5, 7]
  • Determine which has greater standard deviation.
  • Observation: 3, 5, 7 are more spread out than 4, 5, 6, so they have a higher standard deviation.

Steps for Calculation

1. Calculate the Mean
   • Example with [3, 5, 7]:
     • Mean = (3 + 5 + 7) / 3 = 5
2. Apply the Formula
   • For each data point, subtract the mean and square the result.
   • Sum these squared differences.
   • Divide by (n) (for population) or (n-1) (for sample).
   • Take the square root of the result.
3. Example calculation:
   • ( \begin{align*} 3 - 5 &= -2 & \Rightarrow (-2)^2 = 4 \ 5 - 5 &= 0 & \Rightarrow 0^2 = 0 \ 7 - 5 &= 2 & \Rightarrow 2^2 = 4 \ \end{align*} )
   • Sum = 8
   • Standard Deviation [3, 5, 7] = ( \sqrt{\frac{8}{3}} \approx 1.63 )

Variance

• Definition: The square of the standard deviation.
• Example:
  • Standard Deviation = 1.63
  • Variance = 1.63^2 = 2.66 (rounded)
• Formula:
  • ( \text{Variance} = \frac{ \sum{(x_i - \mu)^2} }{n} )

Additional Resources

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